Question

One mole of an ideal monatomic gas undergoes a process described by the equation $PV^{3}=$ constant. The heat capacity of the gas during this process is -

• A. $\frac{3}{2}R$
• B. $\frac{5}{2}R$
• C. $2 \, R$
• D. $R$

Answer 1

The Correct Option is D

The problem describes a thermodynamic process for one mole of an ideal monatomic gas following the relation: PV^{3} = \text{constant}. We are required to find the heat capacity of the gas during this process.

To solve this, we must first understand the relationship given and apply thermodynamic principles to it:

1. The expression PV^{3} = \text{constant} indicates a polytropic process where the polytropic index n = 3.
2. For a polytropic process PV^{n} = \text{constant}, the heat capacity C is given by: C = C_{V} + \frac{R}{1-n}, where C_{V} is the specific heat capacity at constant volume.

For a monatomic ideal gas, C_{V} = \frac{3}{2}R.

3. Substitute n = 3 and C_{V} = \frac{3}{2}R into the equation for C: C = \frac{3}{2}R + \frac{R}{1-3}
4. This simplifies to: C = \frac{3}{2}R - \frac{R}{2}
5. Further simplification gives: C = \frac{3}{2}R - \frac{1}{2}R = R.

Thus, the heat capacity of the gas during this process is R.

Therefore, the correct answer is R.

This result can be justified by understanding that the heat capacity depends on both the specific process the gas undergoes and its specific heat at constant volume. For the given polytropic process, the adjustment for the polytropic index provides us the adjusted heat capacity.